# Full text of "Treatise On Analysis Vol-Ii"

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298 XIV INTEGRATION IN LOCALLY COMPACT GROUPS 3. Let G be a locally compact group, /? a left Haar measure on G. If G =£ {*?}, the algebra L*(G, j8) has zero-divisors =£0. To construct two nonnegligible functions /, g in ^'(G, /?) such that/*# is negligible, we may proceed as follows: (1) The case where G has a compact subgroup H ^ {e}. Take for/a character- istic function q?A and for g a function of the form <psQ — 9?e, where A, B, and s are suitably chosen, and remark that AG(*) = 1 for all x e H. (2) The case G = Z. Show that we may take ' 2/1-1 2/7+1 for all n E Z, and g = /. (3) The general case. Prove first of all that there exists a ^ e in G such that A(fl) = 1. The closure H in G of the subgroup generated by a is then either compact or isomorphic to Z (Section 12.9, Problem 10). In the former case, use the result of case (1); in the latter, take + 00 oo /(/)== 2^ an<pua-n(0> #(0 = ^L Pn'jPani/O n= - oo n- - oo where the set U and the sequences («„), (ft) are suitably chosen with the help of case (2). 4. Let G be a locally compact group, f$ a left Haar measure on G. In order that a subset H of -S?P(G, ft) (1 ^p < H- oo) should have a relatively compact image ft in LP(G, J8), it is necessary and sufficient that the following conditions should be satisfied: (1) ft is bounded in LP(G, /?); (2) for each £ > 0, there exists a compact subset K of G such that Np(/9G-ic) ^ e for all/e H; (3) for each e > 0, there exists a neighborhood V of e in G such that NP((Y(.S)/) —f)^e for all/e H and all s e V. (To prove that these conditions are sufficient, observe that if g G JT(G) and if L is a compact subset of G, then the image, under the mapping/i—*g */, of the set of restrictions to L of functions belonging to H is an equicontinuous subset of . 5. Let G be a locally compact group, j8 a left Haar measure on G, and ft a positive measure on G. Show that if A is a //,-integrable set and B is a universally measurable set in G, the function u : s\—>/x(A n sB) is ^-measurable on G. If moreover B~l is /3-integrable, then so is u and we have (Use the Lebesgue-Fubini theorem and Problem 21 of Section 13.9.) Give an example of a measure ft such that the function s-i— »/x(A n .sB) is not continuous. If ft is a measure with basis /3 and if A is /u-integrable and B is universally measurable, then the function s\— >/x(A n 56) is continuous on G. 6. Let G be a locally compact group, G' a topological group, /a homomorphism of G into G' which is ^-measurable, where ft is a left Haar measure on G, Show that /is continuous. (Observe that if we put #(#) =/(jc~l), there exists a compact non-£- negligible subset K of G such that the restrictions of /and g to K are continuous. Deduce that the restriction of /to K • K"1 is continuous, by using (12.3.8); then apply (14.10.8).) >7/?/'r,\res (/An * (/„ • j8)) converges to (/u,B * (/ • j8)) with respect to ^6